Question

given \\(\overline{bc}\\), describe the error made when rotating \\(b(3, -1)\\) counterclockwise \\(270^0\\) about \\(c(-2, -5)\\).

• step 1: \\((3, -1) \to (1, -6)\\)
• step 2: \\((1, -6) \to (-6, -1)\\)
• step 3: \\((-6, -1) \to (-4, 4)\\)

the image is located at \\(b(-4, 4)\\).

Explanation:

Step 1: Find Vector $\overrightarrow{CB}$

To rotate a point \( B(x,y) \) about a center \( C(h,k) \), first find the vector from \( C \) to \( B \): \( \overrightarrow{CB}=(x - h, y - k) \). Here, \( B(3,-1) \), \( C(-2,-5) \), so \( \overrightarrow{CB}=(3 - (-2), -1 - (-5))=(5,4) \).

Step 2: Recall 270° Counterclockwise Rotation Rule

The rule for a 270° counterclockwise rotation of a vector \( (a,b) \) is \( (a,b)\to(b, -a) \). So, rotating \( \overrightarrow{CB}=(5,4) \) 270° counterclockwise gives \( (4, -5) \).

Step 3: Find Rotated Point \( B' \)

To find \( B' \), add the rotated vector to \( C(h,k) \): \( B'=(h + 4, k - 5)=(-2 + 4, -5 - 5)=(2, -10) \).

Step 4: Analyze Given Steps

• Step 1 Error: The vector \( \overrightarrow{CB} \) was miscalculated. Instead of \( (3 - (-2), -1 - (-5))=(5,4) \), they did \( (3 - 2, -1 - 5)=(1, -6) \) (incorrect subtraction of coordinates).
• Step 2 Error: The 270° rotation rule was misapplied. For a vector \( (a,b) \), 270° counterclockwise is \( (b, -a) \), not the operation done here (which doesn't follow the rule).
• Step 3 Error: After incorrect previous steps, the final addition to \( C \) is also wrong. The correct rotated vector addition was not used.

Answer:

The error occurred in calculating the vector \( \overrightarrow{CB} \) (incorrect subtraction of coordinates), misapplying the 270° counterclockwise rotation rule, and incorrect final point calculation. The correct rotated point should be \( (2, -10) \), not \( (-4, 4) \). The key error was in Step 1 (miscalculating \( \overrightarrow{CB} \)) and Step 2 (wrong rotation rule application).